I would suggest these steps although you can apply this methodology in any order you need to, if for instance you wish to design a vessel of a particular material and size, and wish to known the maximum safe pressure.

1. What operating pressure do you want to be able to achieve?
2. What shape and dimensions is your vessel going to be? E.g. closed ended cylinder with wall thickness t.
3. How can you estimate the stresses that will be seen in the vessel walls? E.g. for a cylinder you can use Lame equations (or thin walled approximations if applicable) for radial and hoop stresses.
4. What safety factor do you want to apply to the design?
5. What material should you thus make it from (taking into account the maximum stress in the vessel walls, the safety factor, and a suitable yield stress for the material)?

That's pretty simplified, but that's approximately how an engineer would approach the problem. Obviously there are many other factors that you should consider, including corrosion of the vessel material (e.g. an iron container holding sea water...), the fatigue life of the vessel if the pressure or temperature varies considerably, variation of ductility with temperature...the list goes on!

From personal experience. If you are designing a large vacuum coating plant to go in the basement - consider the distance between floor and ceiling and the gap between support columns in the corridor it has to go down.
Then factor in the cost of building a shed to put the thing in and how annoyed the users are going to be working on it in winter if the shed isn't heated.

Honestly, if you don't know what you're doing, I would get with someone that does know. Pressure vessels are scary beasts, especially if you are doing something that is going to be installed around people or in a plant where people work.

Stewart was right. ASME BPV code sect VIII is a HUGE tome that covers everything. There are books written on how to read and understand it. Designing a pressure vessel is not simply hoop stress and a factor of safety.

Dennis Moss put out a pretty good book "Pressure Vessel Design Manual." You need to track down a book like that to get a better glimpse as to what you are looking at. You will need to brush up on things like the strength of the heads or caps, weld strength, stresses around openings, relief valve/rupture disc sizing, etc...